Let us define probabilistic latent variable model as a statistical model of observable variables that incorporates unobservable variables. Accordingly, any parametric or nonparametric hierarchical Bayesian models subsumed under the decomposition given by Equations REF  and REF  can be considered as special cases in the general class of probabilistic latent variable models. To formulate a general modeling framework for the family of probabilistic latent variable models, we incorporate a D-dimensional local latent variable vector \(\mathbf {z}_n=(z_1, \hdots ,z_D)\)  for each observation \(\mathbf {z}_n\)  and a M-dimensional global latent variable vector \(\theta =(\theta _1, \hdots ,\theta _M)\)  shared across all observations [1]}. The local latent variable, \(\mathbf {z}_n\) , serves the function of capturing local dependencies within the observation vector \(\mathbf {y}_n\)  (e.g. clusters or common factors) whereas the global latent variable \(\theta \)  captures the dependencies across all, or even a subset, of the \(N\)  observable random vectors. Accordingly, the joint probability distribution for a probabilistic latent variable model can be expressed as such
\(p(\mathbf {y}_1,\cdots , \mathbf {y}_N, \mathbf {z}_1,\cdots , \mathbf {z}_N, \theta )\) 
